Snakes strings balloons and other active contour models
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Snakes, Strings, Balloons and Other Active Contour Models. Goal. Start with image and initial closed curve Evolve curve to lie along “important” features Edges Corners Detected features User input. Applications. Region selection in Photoshop Segmentation of medical images Tracking.

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Snakes, Strings, Balloons and Other Active Contour Models

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Snakes strings balloons and other active contour models

Snakes, Strings, Balloonsand Other Active Contour Models


Snakes strings balloons and other active contour models

Goal

  • Start with image and initial closed curve

  • Evolve curve to lie along “important” features

    • Edges

    • Corners

    • Detected features

    • User input


Applications

Applications

  • Region selection in Photoshop

  • Segmentation of medical images

  • Tracking


Corpus callosum

Corpus Callosum

[Davatzikos and Prince]


Corpus callosum1

Corpus Callosum

[Davatzikos and Prince]


User visible options

User-Visible Options

  • Initialization: user-specified, automatic

  • Curve properties: continuity, smoothness

  • Image features: intensity, edges, corners, …

  • Other forces: hard constraints, springs, attractors, repulsors, …

  • Scale: local, multiresolution, global


Behind the scenes options

Behind-the-Scenes Options

  • Framework: energy minimization, forces acting on curve

  • Curve representation: ideal curve, sampled, spline, implicit function

  • Evolution method: calculus of variations, numerical differential equations, local search


Snakes active contour models

Snakes: Active Contour Models

  • Introduced by Kass, Witkin, and Terzopoulos

  • Framework: energy minimization

    • Bending and stretching curve = more energy

    • Good features = less energy

    • Curve evolves to minimize energy

  • Also “Deformable Contours”


Snakes energy equation

Snakes Energy Equation

  • Parametric representation of curve

  • Energy functional consists of three terms


Internal energy

Internal Energy

  • First term is “membrane” term – minimum energy when curve minimizes length(“soap bubble”)

  • Second term is “thin plate” term – minimum energy when curve is smooth


Internal energy1

Internal Energy

  • Control a and b to vary between extremes

  • Set b to 0 at a point to allow corner

  • Set b to 0 everywhere to let curve follow sharp creases – “strings”


Image energy

Image Energy

  • Variety of terms give different effects

  • For example,minimizes energy at intensity Idesired


Edge attraction

Edge Attraction

  • Gradient-based:

  • Laplacian-based:

  • In both cases, can smooth with Gaussian


Corner attraction

Corner Attraction

  • Can use corner detector we saw last time

  • Alternatively, let q = tan-1 Iy / Ixand let nbe a unit vector perpendicular to the gradient. Then


Constraint forces

Constraint Forces

  • Spring

  • Repulsion


Evolving curve

Evolving Curve

  • Computing forces on v that locally minimize energy gives differential equation for v

    • Euler-Lagrange formula

  • Discretize v: samples (xi, yi)

    • Approximate derivatives with finite differences

  • Iterative numerical solver


Other curve evolution options

Other Curve Evolution Options

  • Exact solution: calculus of variations

  • Write equations directly in terms of forces,not energy

  • Implicit equation solver

  • Search neighborhood of each (xi, yi) for pixel that minimizes energy

    • Shah & Williams paper


Variants on snakes

Variants on Snakes

  • Balloons [Cohen 91]

    • Add inflation force

    • Helps avoid getting stuck on small features


Balloons

Balloons

Balloons

Snakes

[Cohen 91]


Balloons1

Balloons

[Cohen 91]


Other energy or force terms

Other Energy or Force Terms

  • Results of previously-run local algorithms

    • e.g., Canny edge detector output convolvedwith Gaussian

  • Automatically-evolved control points

  • Others…


Brain cortex segmentation

Brain Cortex Segmentation

Add energy term for constant-color regions of a single color

Davatzikos and Prince


Brain cortex segmentation1

Brain Cortex Segmentation

Davatzikos and Prince


Brain cortex segmentation2

Brain Cortex Segmentation

Find features

and add

constraints

Davatzikos and Prince


Brain cortex segmentation3

Brain Cortex Segmentation

Davatzikos and Prince


Scale

Scale

  • In the simplest snakes algorithm, image features only attract locally

  • Greater region of attraction: smooth image

    • Curve might not follow high-frequency detail

  • Multiresolution processing

    • Start with smoothed image to attract curve

    • Finish with unsmoothed image to get details

  • Heuristic for global minimum vs. local minima


Diffusion based methods

Diffusion-Based Methods

  • Another way to attract curve to localized features: vector flow or diffusion methods

  • Example:

    • Find edges using Canny

    • For each point, compute distance tonearest edge

    • Push curve along gradient of distance field


Gradient vector fields

Gradient Vector Fields

Xu and Prince


Gradient vector fields1

Gradient Vector Fields

Simple Snake

With Gradient Vector Field

Xu and Prince


Gradient vector fields2

Gradient Vector Fields

Xu and Prince


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